Quantization of Diffeomorphism-Invariant Theories with Fermions

We extend ideas developed for the loop representation of quantum gravity to diffeomorphism-invariant gauge theories coupled to fermions. Let P → Σ be a principal G-bundle over space and let F be a vector bundle associated to P whose fiber is a sum of continuous unitary irreducible representations of the compact connected gauge group G, each representation appearing together with its dual. We consider theories whose classical configuration space is A × F , where A is the space of connections on P and F is the space of sections of F , regarded as a collection of Grassmannvalued fermionic fields. We construct the ‘quantum configuration space A × F as a completion of A × F . Using this we construct a Hilbert space L(A×F) for the quantum theory on which all automorphisms of P act as unitary operators, and determine an explicit ‘spin network basis’ of the subspace L((A × F)/G) consisting of gauge-invariant states. We represent observables constructed from holonomies of the connection along paths together with fermionic fields and their conjugate momenta as operators on L((A× F)/G). We also construct a Hilbert space Hdiff of diffeomorphism-invariant states using the group averaging procedure of Ashtekar, Lewandowski, Marolf, Mourão and Thiemann.